Different traditions have more nuance in how they understand the trinity, but the most relevant and agreed upon part of that definition is: God is three 'persons', Father, Son, Holy Spirit, and each 'person' is clearly distinguishable from the other two, and all three exist simultaneously.
Binitarianism – belief that God is a 'binitarian' being – is effectively identical to trinitarianism, with one exception: the holy spirit is not considered to be a 'person'.
So where the bible speaks of the Father and/or the Son, binitarians typically agree with trinitarians on how to interpret those texts. Disagreement comes down to how to interpret parts of the bible where the holy spirit is mentioned.
I don't think it is necessary here to review every possible verse of relevance, but I will instead mention a few key texts that shape the binitarian approach to the bible:
Focus is placed on a verse such as 1 Corinthians 8.6, where Paul describes Christian faith being centered around 'one God, the Father' and 'one Lord, Jesus Christ', with no mention of the holy spirit.
First John 1.3 says 'our fellowship is with the Father and with his Son Jesus Christ' (ESV). Again, binitarians note the lack of reference to the holy spirit as a 'person' we share fellowship with.
Revelation 3.21, 5.6, 7.17, and 22.1,3 portray Jesus (the Lamb) and God sharing a single throne, but no portrayal is made of the holy spirit also sharing this throne.
Drawing from both the ideas represented in both 2 and 3 above, further emphasis is placed on other texts that focus on God and Jesus, with no reference to the holy spirit when we would otherwise 'expect' to find such a reference (e.g. Romans 10.9, 1 Corinthians 15.20-28, Hebrews 1.1-4).
Most of the epistles begin with the author identifying himself, identifying his intended readership, and greeting where blessings are invoked. All of the greetings invoke blessing from God the Father, or from God and Jesus. E.g., Paul often greets his readers with, 'Grace and peace to you from God our Father and the Lord Jesus Christ'. Binitarians take the lack of mention of the holy spirit as an indication that the spirit is not a distinct 'person' the way we understand God and Jesus to be.
Additional points made are that, within the new testament, we find examples of prayer, worship, and praise directed toward God and Jesus, but no prayer, worship, or praise directed toward the holy spirit.
Binitarianism ultimately sees theology, soteriology, eschatology, etc., of the new testament taking shape around two distinct persons, God and Jesus. The holy spirit is understood as the singular spirit of God and Jesus (e.g. Romans 8.9) but not a distinct person, analogous to how a human's spirit is not a distinct person.
However, depending on some contexts, the term 'holy spirit' is not understood by binitarians to have a systematic definition. In some contexts the term is understood as referring to God's spirit (e.g. the parallelism between Matthew 10.20 and Luke 12.12), but in other contexts it may be understood as something more general (e.g. Psalm 51.11 as a poetic parallel to God's 'presence', or Luke 1.35 as a parallel to 'the power of the Most High').